Subscribe! http://www.freemathvideos.com Welcome, ladies and gentlemen. So what I'd like to do is show you how to determine the vertices, the center, the co-vertices, as well as the foci of an ellipse.
So basically, in the last video, we did the exact same thing, except the center was at the origin, (0,0) so we already had that kind of knocked out of the way. In this example though, we need to figure out what the center is. And I wrote in the center. And it's just like our videos for a circle. The center is going to be (H,K).
Basically whatever we're subtracting from our x and subtracting from our k are going to be our h and our k. Now, I don't really want to get too in-depth with this. But again, if you haven't watched any of those previous videos, remember we can rewrite a subtraction problem as x minus negative 3. Because remember it's x minus h, right? Well x minus negative 3 is the same thing as x plus 3.
So if you look at this, what are you subtracting from 3? It's x minus h, x minus negative 3. That means, h is equal to negative 3. So in this example, we can say that the center is at the opposite of 3, which is negative 3, and the opposite of 1, which is negative 1.
OK. So once we kind of knock out the center, the next thing we want to do is figure out our a, b, and c. Remember the a represents-- or actually I wrote in what a represents, b represents and c represents. Remember, a is the distance from the center to your vertices, which represent your major axis. Or your vertices are the endpoints of your major axis. Remember the major axis is always larger than your minor axis.
So therefore a squared is always going to be larger than b-squared. Because b remembers the distance from the center to your co-vertices, and the co-vertices are the endpoints of your minor axis. Major axis is always larger than minor axis. a is the distance from the center of the vertices. b is the distance from the center of the co-vertices. Therefore a is always going to be larger than b for an ellipse.
So, we have-- if you look at your a-squared and your b-squareds as your denominators-- so we have two. We just need to decide which one is a squared, which one's b squared. Well, remember, a squared is always larger than b squared. So therefore I can make the determination that a squared is equal to 9 and b squared is equal to 4. Excuse me. Excuse me. OK.
So therefore a is equal to 3 and b is equal to 2. Now again, to be able to determine what c squared is, we can use the relationship. So I can say that c squared is equal to a squared minus b squared. So c squared is equal to 9 minus 4. c squared is equal to 5. Take the square root on both sides. C is equal to the square root of 5.
Now, a lot of times to kind of makes sense of this, again we need to plot the information. And this is the exact same thing I did for when our center was at (0,0). Actually, you know what? Let me write this in red, so we can differentiate all the information. Center is at (-3,-1). OK.
So we have a center at (-3, -1) so I'm going to go -3, 1, 2, 3, down 1. All right. Now, I prefer to plot the information rather than doing some formula. That kind of makes sense because once I can plot the information, I can determine where everything is at. Or I can determine what those values are that I'm going to write down.
Now remember, a squared was 9, right, because a squared was always larger than b squared. If a square is under the x, that means my major axis is horizontal. So a lot of times what I like to do is, I write in just my major axis. I write it is a nice dotted line and I just write down major axis. This lets me not forget that the major axis is horizontal.
And that's very important because, not only do the vertices lie on the major axis, the foci lie on the major axis. And the co-vertices lie on the minor axis, which is perpendicular to the major axis. And they intersect at the center. So, even though these are not really a part of the graph, I'm not really graphing them. I'm just plotting the information so I know where it is. So I'm just going to plot in the minor axis, which is perpendicular to the major axis. OK.
So, to find my vertices, the distance from the vertici to the center is a, which in this case is 3. So I go to my center, and I say, all right, it needs to go along the major axis-- 3 units to the right, and 3 units to the left. Because remember an ellipse has end-points, to the left and to the right.
So, what I'm going to do is from this point, I'm just going to go to the right 3 units, 1, 2, 3, and to the left three units, 1, 2, 3. OK. You can see me. Good. Now, I basically just plot-- or, actually, let's do everything. I'll go ahead and label those as my vertices. Now along that same path I'm going to do my co-vertices, which is the square root of 5.
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